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Sia ? 6 years, 6 months ago
By Euclid’s division lemma, we have
a = bq + r; 0 ≤ r < b
For a = n and b = 5, we have
n = 5q + r …(i)
Where q is an integer
and 0 ≤ r < 5, i.e. r = 0, 1, 2, 3, 4.
Putting r = 0 in (i), we get
n = 5q
⇒ n is divisible by 5.
n + 4 = 5q + 4
⇒ n + 4 is not divisible by 5.
n + 8 = 5q + 8
⇒ n + 8 is not divisible by 5.
n + 12 = 5q + 12
⇒ n + 12 is not divisible by 5.
n + 16 = 5q + 16
⇒ n + 16 is not divisible by 5.
Putting r = 1 in (i), we get
n = 5q + 1
⇒ n is not divisible by 5.
n + 4 = 5q + 5 = 5(q + 1)
⇒ n + 4 is divisible by 5.
n + 8 = 5q + 9
⇒ n + 8 is not divisible by 5.
n + 12 = 5q + 13
⇒ n + 12 is not divisible by 5.
n + 16 = 5q + 17
⇒ n + 16 is not divisible by 5.
Putting r = 2 in (i), we get
n = 5q + 2
⇒ n is not divisible by 5.
n + 4 = 5q + 9
⇒ n + 4 is not divisible by 5.
n + 8 = 5q + 10 = 5(q + 2)
⇒ n + 8 is divisible by 5.
n + 12 = 5q + 14
⇒ n + 12 is not divisible by 5.
n + 16 = 5q + 18
⇒ n + 16 is not divisible by 5.
Putting r = 3 in (i), we get
n = 5q + 3
⇒ n is not divisible by 5.
n + 4 = 5q + 7
⇒ n + 4 is not divisible by 5.
n + 8 = 5q + 11
⇒ n + 8 is not divisible by 5.
n + 12 = 5q + 15 = 5(q + 3)
⇒ n + 12 is divisible by 5.
n + 16 = 5q + 19
⇒ n + 16 is not divisible by 5.
Putting r = 4 in (i), we get
n = 5q + 4
⇒ n is not divisible by 5.
n + 4 = 5q + 8
⇒ n + 4 is not divisible by 5.
n + 8 = 5q + 12
⇒ n + 8 is not divisible by 5.
n + 12 = 5q + 16
⇒ n + 12 is not divisible by 5.
n + 16 = 5q + 20 = 5(q + 4)
⇒ n + 16 is divisible by 5.
Thus for each value of r such that 0 ≤ r < 5 only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.
Posted by Amrita Singh 6 years, 6 months ago
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Sia ? 6 years, 6 months ago
Since, any odd positive integer n is of the form 4m + 1 or 4m + 3.
if n = 4m + 1
n2 = (4m + 1)2
= 16m2 + 8m + 1
= 8(2m2 + m) + 1
So n2 = 8q + 1 ........... (i)) (where q = 2m2 + m is a positive integer)
If n = (4m + 3)
n2 = (4m + 3)2
= 16m2 + 24m + 9
= 8(2m2 + 3m + 1) + 1
So n2 = 8q + 1 ...... (ii) (where q = 2m2 + 3m + 1 is a positive integer)
From (i) and (ii) we conclude that the square of an odd positive integer is of the form 8q + 1, for some integer q.
Posted by Rahul Kumar Kushwaha 7 years, 8 months ago
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Milind Patel 7 years, 8 months ago
Posted by Heikhomba Ongnam 6 years, 6 months ago
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Sia ? 6 years, 6 months ago

It is given that:
{tex}\frac { A B } { P Q } = \frac { B C } { Q R } = \frac { A D } { P M }{/tex}
{tex}\Rightarrow \quad \frac { A B } { P Q } = \frac { A D } { P M }{/tex}{tex}= \frac { B C } { Q R } = \frac { \frac { 1 } { 2 } B C } { \frac { 1 } { 2 } Q R } = \frac { B D } { Q M }{/tex} ....................................(i)
In {tex}\triangle ABD{/tex} and {tex}\triangle PQM{/tex}, we have
{tex}\frac { A B } { P Q } = \frac { A D } { P M } = \frac { B D } { Q M }{/tex} [from(i)]
{tex}\therefore \quad \triangle A B D \sim \triangle P Q M{/tex} [by SSS-similarity criteria].
And also, {tex}\angle B = \angle Q{/tex} [corresponding angles of similar triangles are equal].
Now, in {tex}\triangle ABC{/tex} and {tex}\triangle PQR{/tex}, we have
{tex}\angle B = \angle Q{/tex} [proved above]
and {tex}\frac { A B } { P Q } = \frac { B D } { Q M }{/tex} [from(i)].
{tex}\therefore \quad \triangle A B C \sim \triangle P Q R{/tex} [by SAS-similarity criteria].
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Check Syllabus here : <a href="https://mycbseguide.com/cbse-syllabus.html">https://mycbseguide.com/cbse-syllabus.html</a>
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Milind Patel 7 years, 8 months ago
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